Spectral properties with the difference between topological indices in graphs (Q827206)

Summary: Let \(G\) be a graph of order \(n\) with vertices labeled as \(v_1, v_2,\dots, v_n\). Let \(d_i\) be the degree of the vertex \(v_i\), for \(i=1,2,\dots,n\). The difference adjacency matrix of \(G\) is the square matrix of order \(n\) whose \((i,j)\) entry is equal to \((\sqrt{ d_i + d_j - 2} - 1)/( \sqrt{d_i d_j})\) if the vertices \(v_i\) and \(v_j\) of \(G\) are adjacent or \((v_i v_j \in E (G))\) and zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied. We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph.